PoS(BDMH2004)089
Chiara Tonini , Paolo Salucci
SISSA, Italy
E-mail:tonini@sissa.it,salucci@sissa.it
We construct a generic mass model of a disk galaxy by combining two mass components, an exponential thin stellar disk and a pseudo-isothermal dark halo. We construct its generic rotation curve, with two free parameters linked to its shape and one to its amplitude, namely the halo core radius, the fractional amount of Dark Matter and the total mass at one disk scale lenght.
We conduct a statystical analysis of the fitting procedure performed with the model on a family of rotation curves, and we conclude that an observed rotation curve fulfilling well defined quality requirements can be uniquely and properly decomposed in its dark and luminous components; through a suitable mass modelling together with a maximum-likelihood fitting method, in fact, the disentangling of the mass components is accomplished with a very high resolution.
Baryons in Dark Matter Halos 5-9 October 2004
Novigrad, Croatia
PoS(BDMH2004)089
The rotation curves of spiral galaxies are determined by the mass distribution, including its luminouscom-ponent (the disk) and its dark comcom-ponent (the halo). While the radial distribution of the luminous matter is observed for spiral disks, the disk mass itself is unknown. Regarding the Dark Matter, both the amount and the distribution are unknown, and since these physical quantities affect the rotation velocity profile in a different way at different radii, the study of the rotation curves becomes a very important tool to investigate the structure of galaxies.
But how well does a rotation curve yield the true mass model of a galaxy? Moreover, is the information we obtain from the observational data unique, and what is the resolution of our result? In other words, can we distinguish between different Dark Matter amounts and distributions by analyzing a rotation curve? We model the rotation curves of a galaxy from the distribution of the luminous and Dark Matter. We assume a fixed mass density distribution of the Dark Matter, i.e. a pseudo-isothermal (PI) halo, which agrees with a number of observations ([6]; notice that the Universal Rotation Curve and the individual RCs are well fitted by a PI
disk model, [1]; [3]), and we investigate the capability of the model to fit simulated “observed”
rotation curves. In section (1) we present the mass model, while in section (2) we investigate the possibility of uniquely deriving, from an high-quality rotation curve, the mass distribution. In section (3) we finally conclude.
1. The model
The total mass of a disk galaxy can be divided into two independent components, namely the luminous disk and the Dark Matter halo (neglecting the bulge is irrelevant in our study). These components contribute separately to the total rotation velocity:
V2 ˜r V 2 d ˜r Vh2 ˜r (1.1)
For the luminous matter, we consider the stellar disk, whose surface density profile is found as: Σ
r Σ0e
R RD
(1.2) whereΣ0is the central surface density, and RDis the disk scale lenght. The disk component of the velocity
is [2]: V2 d ˜r GMD RD ν ˜r (1.3)
in which MD is the disk mass andν
˜r ˜r 2 2 I0K0 I1K1 ˜r
2, with I0K0 and I1K1 being the Bessel
functions of order 0 and 1, evaluated at ˜r
2. The contribution of the luminous matter to the total rotation velocity is determined by two parameters: the disk scale lenght RD, measured in each galaxy, and the total
mass MD, that can be obtained, at least in principle, from the disk luminosity, but since its determination is
affected by large uncertainties, it is usually considered as an unknown quantity. The DM density distribution of pseudo-isothermal (PI) halos is given by:
ρ r ρ0 1 r R C 2 (1.4)
For convenience we define: ˜r r
R
Das the galactocentric distance, and ˜RC RC
R
Das the core radius of
the halo, both in units of the disk scale lenght. We take ˜RCas a free parameter and we consider its possible
variation range between 0 and infinity. ˜RC 1 implies a mass distribution very similar to that of the NFW
profile (see subsection 2.2). On the other hand, cases with ˜RC 3 are scarcely distinguishable from one
another, since most of the observational data do not reach regions of disks beyond ˜r 3.
The halo velocity component is:
Vh2
˜r
GMH
˜r
˜r RD (1.5)
and it has two free parameters, the central halo density and the halo characteristic scale lenght, that fully determine the DM distribution. The mass of the dark halo as a function of the normalized radius is obtained
PoS(BDMH2004)089
as: MH˜r 4πGρ0 ˜R 3 CR3D˜r ˜RC tan 1 ˜r ˜RC , and we define:λ ˜r ˜R 3 C ˜r ˜r ˜RC tan 1 ˜r ˜RC . It isuseful to perform the following transformation: α GMD RDV12 β 4πGρ0R3D RDV12 λ1 ˜R 3 C ˜R1 C tan 1! 1 ˜RC"$#
The parametersα andβare directly proportional to the fraction, at RD, of the luminous and dark mass
respectively. We define V2
1 V
2
˜r
1%. The radial profile of the normalized rotation velocity is then given
by: V2 ˜r V2 1 α ν ˜r '& β λ ˜r (1.6)
α,βandλdetermine the shape of the curve, while V2
1 determines its amplitude. Let’s stress that, as
con-cerning the results of this section, the actual value of the constant V1is irrelevant since, as a normalization
parameter, it doesn’t affect the RC shape. From (1.6) it follows that:α ν1& β λ1 1, whereν1 ν
˜r
1.
We notice thatαν1andβλ1represent the fractional contributions to the total (normalized) velocity at RDdue
to the luminous and Dark Matter respectively; for simplicity we define fDM βλ1.
We now rewrite the radial profile of the normalizated rotation velocity as
V2 ˜r V2 1 1 fDM ν˜r ν1 & fDM λ ˜r λ1 (1.7)
under the following ranges of variations: 0( fDM ( 1 and 0( ˜RC ( 3. In this way, we split the original
three free parameters into a 2& 1 combination: 2 determine the RC profile and 1 sets the RC amplitude. We
allow the free parameters related with the profile to vary over a very wide range of values, and we plot the corresponding curves as functions of radius ˜r.
2. Disentangling a high-quality RC
We now estimate the precision with which we can resolve the luminous and the Dark Matter distribution from the rotation curve. We build a family of 25 reference curves that simulate “observed” RCs of real galaxies, each one consisting of 25 data points, evenly spread between 0 and 4 disk scale lenghts, and is characterized by a set of two parameters in the ensemble: fDM *) 0+1
0+3 0+5 0+7 0+9,$- ˜RC .) 0+1 1+0 2+0 3+0 4+0, .
To each point of these curves we assign “observational” errors of the order to those of high-quality published RCs [5]:εV 0+02 is the averaged uncertainty in V
˜r
, andεD 0+05 in the RC slope.
We investigate each of the “observed” RCs with a mass model, fitting the data to obtain their structural parameters. Regarding the parameter V1, if the observed RC is of high quality, then it’s possible to set its
value directly from the data. In this case, V1 V
1
0/
O
10
2
. Let us caution that this cannot always
be done: we define a high-quality RC as a set of kinematical data for which such a parameter reduction is possible. Therefore, the reference “observed” RCs are fitted with the following mass model (see eq. 1.7):
V2 mod˜r V 2 1 1 fDM γL ˜r 1& fDMγDM ˜r; ˜RC 2 (2.1) whereγL˜r ν ˜r 3 ν1is known, andγDM ˜r; ˜RC λ ˜r; ˜RC 4 λ1 ˜RC .
The measure of the distance between a model curve and an “observed” RC is given by a likelihood parameter, that we define as the sum of theχ2computed on the velocity and on the RC slope:
χ2 TOT χ 2 V 5& χ 2 D (2.2)
The need of considering the contribution of the gradient of the velocity in the computation of the likelihood parameter is well explained by the examples shown in figure (1). In the top panels we show the value of the χ2
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1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 -1 -0.5 0 0.5 1Figure 1:Top panels:χ26
V7 for 3 reference curves, as a function of the model parameters: ˜RC
8:90
;2<4= (x-axis) and fDM
8:90
;05<1=
(y-axis). The “true” parameters are: ( ˜RC> 1
;0<fDM > 0 ;3) (left), ( ˜RC > 2 < fDM > 0 ;5) (middle), ( ˜RC > 3 <fDM > 0 ;7) (right). Bottom panels:χ2
T OTfor the same cases.
contour ofχ2
V includes also values of the parameters far from the “true” ones, unveiling a certain inability
of the model to disentangle the mass components. With the information given by the gradient of the circular velocity and by theχ2
TOT as defined in equation (2.2), the degeneracy of the different model curves largely
disappears. In the bottom panels we show the resolution reached now: we find that theχ2
TOT rises sharply to
a value of? 10
2in an interval of∆f
DM ? 0@05 and of∆˜RC
A 0
@25.
For each one of the 25 “observed” reference curves, we perform the fit with the model given by equa-tion (2.1), considering a very large range for the values of its parameters. We quantify the “success” of the fit by the computation of χ2
TOT for each model curve considered. We now perform a general
anal-ysis of the behaviour of χ2
TOT, computed for 10.000 couples reference “observed” RC - model curve.
It is worth to consider χ2
TOT as a function of the following suitable “distance” in the paramater space:
DBDC∆ 2 C ˜RC E 4 FHG ∆ 2 C fDMF3F 1I2 , with∆CxF J xmod
K xobs (notice that, since the range of variation of RC
is four times that of fDM, we normalize the former by a factor of 4 to make the two contributions to D
comparable). The result is shown in figure (2a). The red circles represent cases in which the distance is dominated by the variation of the core radius (∆2
C ˜RC
E 4
FL ∆
2
C fDMF), the blue triangles cases in which the
distance is mainly due to the variation of the amount of Dark Matter. The vertical line defines a distance in the parameter space of DJ 0@25. The points in the range DM 0@25 and laying below the 3σlimit, although
representing cases in which the model fails in fitting uniquely a RC, can be considered irrelevant and will not be counted in the statystics: in fact, the parameters defining the curves are so close to each other, that even if a RC has multiple fits, they differ one from the other by a negligible amount.
The horizontal lines represent the limit value ofχ2
TOT corresponding to 3σand 1σ. Most of the points lay
over the 3σlimit, indicating that the two curves of each couple are well resolved and therefore distinguish-able. In the interval 1K 3σwe find about 0
@9% of the total cases, and below the 1σlimit about 0@23%. The
cases below the 1σlimit have to be considered as failures of the model in uniquely distinguishing the Dark Matter distribution, and those below the 3σlimit as partial failures: therefore, studying our simulated “ob-served” RCs we find that there is a total possibility of only about 1% not to resolve (completely or partially) the mass model, with the adoptedεV,εD: these errors are typical of about the top 50% of the RCs observed
today, for which, through the application of this procedure, there are very good possibilities to disentangle the mass distribution. However, it is possible to obtain an observed RC with an error even smaller than the
PoS(BDMH2004)089
a) b)
Figure 2:χ2
TOTof the 25 reference “observed” curves mapped with 400 model curves, as a function of the distance in the parameter
space. Red points (circles) are for distances dominated by the variation of ˜RC, blue points (triangles) are for distances dominated by the
variation of fDM. The straight lines represent the 1σand 3σlimits. Panel a)εVN 0 O02 andεD N 0 O05. Panel b)ε N 0 O01 andεD N 0 O03.
one we adopted: for about the top 10% of the RCs available today we can consider observational errors ofεV P 0
Q01 andεD
P 0
Q03, and the number of these curves is going to increase in the future. The same
analysis performed with these errors yields aχ2
TOTvs D relation as shown in (2b). Now we have only about
the 0Q15% of cases in the interval 1R 3σ, and only the 0Q02% is below 1σ; the worst case among these
sets the lower limit for the resolution of our model: S∆S fDM
T3TMAX P 0
Q25 andS∆RC
TMAX P 1
Q2RD, which is
still a reasonable uncertainty. Therefore, considered the very small amount of cases below the 3σlimit, we can state that the average resolution that we reach corresponds to the step with which we select the model parameters, i.e. S∆S fDM T3T P 0 Q05 andS∆RC T P 0 Q2RD.
3. Conclusions
We performed a mass modelling of the rotation curve of a spiral galaxy, representing it as a function of two parameters that govern its shape and one that governs its amplitude, namely the dark halo core radius, the Dark Matter contribution to the total rotation velocity and the total velocity at one disk scale lenght. Under the hypothesis of investigating a high-quality RC, we show that the model is able to disentangle the mass components of the galaxy and to determine the DM distribution.
Our results refer to the adopted halo profile, without any attempt to investigate whether this is the actual profile of DM in spirals. Nevertheless, the same analysis could be conducted with the NFW halo [4], for instance, and in this case the same results would apply, since the NFW profile, having one free parameter less than the PI case and featuring a known radial behaviour of the velocity profile, is easier to reconstruct from a rotation curve.
References
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[3] Gentile, G., Salucci, P., Klein, U., Vergani, D., Kalberla, P. 2004, MNRAS, 351, 903G [4] Navarro, J. F., Frenk, C. S., White, S. D. M. 1997, ApJ, 490, 493
[5] Persic, M., Salucci, P. 1995, ApJS, 99, 501P [6] Persic, M., Salucci, P., Stel, F. 1996, MNRAS, 281...27P
